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All the random variables below are defined on the same probabiluty space $(\Omega, \mathcal{F}, \mathbb{P})$.

Consider the following model

$$ Y\equiv 1\{\epsilon > \beta_0+\beta_1X\} $$ where

  • $1$ is an indicator function equal to $1$ if the condition inside is satisfied and zero otherwise

  • $Y$ is a random variable with support $\mathcal{Y}\equiv \{0,1\}$

  • $X$ is a random variable with support $\mathcal{X}\equiv \{0,1\}$ (treatment)

  • $\epsilon$ is a random variable with support $\mathbb{R}$

  • $\beta_0, \beta_1$ are scalar parameters

A source I am considering states that the average treatment effect (ATE) is

$$ \begin{aligned} \text{ATE} & = P_{\epsilon}(\epsilon > \beta_0+\beta_1)- P_{\epsilon}(\epsilon > \beta_0) \end{aligned} $$ without specifying which probability is $P_{\epsilon}$.

Question: Is the probability used to compute the ATE, $P_{\epsilon}$, conditional on $X$ or unconditional?


I am tempted to say that it should be conditional; in other words

$$ \begin{aligned} \text{ATE} & \equiv E(Y|X=1)-E(Y|X=0)= \mathbb{P}(\epsilon > \beta_0+\beta_1|X=1)- \mathbb{P}(\epsilon > \beta_0|X=0) \end{aligned} $$

However, then the book claims that

$$ \begin{aligned} \text{ATE} & = P_{\epsilon}(\epsilon > \beta_0+\beta_1)- P_{\epsilon}(\epsilon > \beta_0) \\ &= \sum_{x\in \{0,1\}} \mathbb{P}(\epsilon > \beta_0+\beta_1|X=x) \mathbb{P}(X=x)\\ &-\sum_{x\in \{0,1\}} \mathbb{P}(\epsilon > \beta_0|X=x) \mathbb{P}(X=x) \end{aligned} $$ which seems to be an application of he law of iterated expectations making sense only if $P_{\epsilon}$ is not conditional on $X$.

Any hint?

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  • $\begingroup$ From which book is this? $\endgroup$ – Julian Schuessler Jul 25 '18 at 8:52
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The ATE is defined as $ATE = \mathbb{P}(Y = 1|do(X =1)) - \mathbb{P}(Y =1 |do(X=0))$. When you intervene on $X$, you force it to be the value you want. Thus,
$$\mathbb{P}(Y= 1|do(X =1)) = \mathbb{P}(1(\epsilon>\beta_0 + \beta_1(1)) = 1) = \mathbb{P}(\epsilon>\beta_0 + \beta_1)$$

and, analogously,

$$\mathbb{P}(Y=1|do(X=0))= \mathbb{P}(1(\epsilon>\beta_0 + \beta_1(0)) = 1) = \mathbb{P}(\epsilon > \beta_0)$$

Then,

$$ATE = \mathbb{P}(\epsilon>\beta_0 + \beta_1) - \mathbb{P}(\epsilon > \beta_0)$$

Your confusion stems from defining ATE in terms of the difference in conditional expectations, an observational quantity. ATE is a structural/interventional quantity, not an observational quantity. They are only equal under certain conditions, such as $\epsilon \perp X$. The task of finding an observational expression that is equivalent to an interventional expression is what we call identification.

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